Universality in a 2-Component Fermi System at Finite Temperature

Thermodynamic properties of a Fermi system close to the unitarity limit, where the 2-body scattering length $a$ approaches $\pm \infty$, are studied in the high temperature Boltzmann regime. For dilute systems the virial expansion coefficients in the Boltzmann regime are expected, from general arguments, to be universal. A model independent finite temperature $T$ calculation of the third virial coefficient $b_3(T)$ is presented. At the unitarity limit, $b_3^{\infty }\approx 1.11$ is a universal number. The energy density up to the third virial expansion is derived. These calculations are of interest in dilute neutron matter and could be tested in current atomic experiments on dilute Fermi gases near the Feshbach resonance.

Figure 1

Plot of $b_2^{(2)}$ and its derivative $\Delta b_2\equiv T{\partial }_T{b}_2^{(2)}$ for ${}^6\mathrm{Li}$ as a function of the inverse scattering length $\gamma =1/a$ in eV. $1\mathrm{eV}\approx 5.06\times {10}^{-4}{\AA }^{-1}$ in natural units ($\hslash =1=c$). Solid, long-dashed, dot-dashed, and short-dashed curves correspond to temperatures of $6\mu \mathrm{K}$, $8.4\mu \mathrm{K}$, $11.76\mu \mathrm{K}$, and $16.46\mu \mathrm{K}$, respectively.

Figure 2

The pressure $P$ from 1- and 2-body physics. Solid lines are fermion fields. Each particle-particle loop is order $g\sqrt{MT}\sim a\sqrt{MT}$ and summed to all orders to form the dressed dimer propagator, represented by the double lines.

Figure 3

3-body contribution to the pressure $P_3^{(3)}$.

Figure 4

Plot of ${\overline{b}}_3$ and its derivative $\Delta b_3\equiv T{\partial }_T{\overline{b}}_3$ for ${}^6\mathrm{Li}$ as a function of $\gamma =1/a$ in eV. The different curves are for the same set of temperatures as in Fig. 1 and use the same notations.

Figure 5

The second (solid curve) and the third order (dashed curve) virial contribution in the $n{\lambda }^3/2$ expansion in Eq. (11).